Stress-strain and limiting state of spoked flywheels formed from composite materials. Report 1. Computational relationships
The purpose of the present study is to obtain a rather simple solution for this problem which is convenient for practical engineering computations, and to evaluate its discrepancy with the accurate solution, and also its correspondence with experimental data on the limiting bearing capacity for two...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
1985
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| Цитувати: | Stress-strain and limiting state of spoked flywheels formed from composite materials. Report 1. Computational relationships / V.M. Leshchenko, B.D. Kosov, I.A. Kozlov // Проблемы прочности. — 1985. — № 8. — С. 1151-1157. — Бібліогр.: 1 назв. — англ. |
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irk-123456789-1828922022-01-23T01:27:13Z Stress-strain and limiting state of spoked flywheels formed from composite materials. Report 1. Computational relationships Leshchenko, V.M. Kosov, B.D. Kozlov, I.A. Scientific-technical section The purpose of the present study is to obtain a rather simple solution for this problem which is convenient for practical engineering computations, and to evaluate its discrepancy with the accurate solution, and also its correspondence with experimental data on the limiting bearing capacity for two flywheel models of various composite materials. 1985 Article Stress-strain and limiting state of spoked flywheels formed from composite materials. Report 1. Computational relationships / V.M. Leshchenko, B.D. Kosov, I.A. Kozlov // Проблемы прочности. — 1985. — № 8. — С. 1151-1157. — Бібліогр.: 1 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/182892 678.2:629.7 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Scientific-technical section Scientific-technical section |
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Scientific-technical section Scientific-technical section Leshchenko, V.M. Kosov, B.D. Kozlov, I.A. Stress-strain and limiting state of spoked flywheels formed from composite materials. Report 1. Computational relationships Проблемы прочности |
| description |
The purpose of the present study is to obtain a rather simple solution for this problem which is convenient for practical engineering computations, and to evaluate its discrepancy with the accurate solution, and also its correspondence with experimental data on the limiting bearing capacity for two flywheel models of various composite materials. |
| format |
Article |
| author |
Leshchenko, V.M. Kosov, B.D. Kozlov, I.A. |
| author_facet |
Leshchenko, V.M. Kosov, B.D. Kozlov, I.A. |
| author_sort |
Leshchenko, V.M. |
| title |
Stress-strain and limiting state of spoked flywheels formed from composite materials. Report 1. Computational relationships |
| title_short |
Stress-strain and limiting state of spoked flywheels formed from composite materials. Report 1. Computational relationships |
| title_full |
Stress-strain and limiting state of spoked flywheels formed from composite materials. Report 1. Computational relationships |
| title_fullStr |
Stress-strain and limiting state of spoked flywheels formed from composite materials. Report 1. Computational relationships |
| title_full_unstemmed |
Stress-strain and limiting state of spoked flywheels formed from composite materials. Report 1. Computational relationships |
| title_sort |
stress-strain and limiting state of spoked flywheels formed from composite materials. report 1. computational relationships |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
1985 |
| topic_facet |
Scientific-technical section |
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http://dspace.nbuv.gov.ua/handle/123456789/182892 |
| citation_txt |
Stress-strain and limiting state of spoked flywheels formed from composite materials. Report 1. Computational relationships / V.M. Leshchenko, B.D. Kosov, I.A. Kozlov // Проблемы прочности. — 1985. — № 8. — С. 1151-1157. — Бібліогр.: 1 назв. — англ. |
| series |
Проблемы прочности |
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| first_indexed |
2025-07-16T02:08:59Z |
| last_indexed |
2025-07-16T02:08:59Z |
| _version_ |
1837767584636731392 |
| fulltext |
i,
2.
3.
4.
5.
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7.
8.
LITERATURE CITED
D. P. H. Hasselman, "Figures-of-merlt for the thermal stress resistance of high tempera-
ture brittle materials: a review," Ceramurgia Int., ~, No. 4, 147-150 (1978).
G. S. Pisarenko, V. N. Rudenko, G. N. Tret'yachenko, and V. T. Troshchenko, Strength
of Materials at High Temperatures [in Russian], Naukova Dumka, Kiev (1966).
G. A. Gogotsi, Inelasticity o~ Ceramics and Refractories [in Russian], Inst. Probl.
Prochn. Akad. Nauk Ukr. SSSR, Kiev (1982).
G. A. Prantskyavichyus, "Experimental and analytical investigation of the thermal
failure of the oxides of high-refractory materials," Author's Abstract of Candidate's
Dissertation, Technical Sciences (1968), p. 29.
I. I. Nemets, "Analysis of the relation between the thermal-stability characteristics,
deformability, strength, and thermal expansion of ceramics," Probl. Prochn., No. ii,
48-51 (1981).
K. A. Kazakyavlchyus and A. I. Yanulyavlchyus, Laws Governing the Thermal Failure of
Prismatic Solids [in Russian], Mokslas, Vilnius (1981).
B. C. Gatewood, Thermal Stresses, McGraw-Hill (1957).
E. Ya. Litovskii and N. A. Puchkelevich, Thermophysical Properties of Refractories [in
Russian], Metallurigya, Moscow (1982).
STRESS--STRAIN AND LIMITING STATE OF SPOKED FLYWHEELS
FORMED FROM COMPOSITE MATERIALS.
REPORT I. COMPUTATIONAL RELATIONSHIPS
V. M. Leshchenko, B. D. Kosov,
and I. A. Kozlov
UDC 678.2:629.7
Computation of the stress-straln state of a flywheel built from composite materials in
the form of an energy-storlng element -- a rim connected with spokes using a chorded star-
shaped polygon (Fig. I) -- is performed in a rather complete statement with allowance for the
interaction between spokes at the points of their intersection. In addition to this, the
effect of this interaction on the adequacy of the solution, which has been found to be rather
cumbersome, has not been evaluated.
The purpose of the present study is to obtain a rather simple solution for this problem
which is convenient for practical engineering computations, and to evaluate its discrepancy
with the accurate [i] solution, and also its correspondence with experimental data on the
limiting bearing capacity for two flywheel models of various composite materials.
We computed the stress--strain state of a flywheel with consideration given to the Joint
deformation of the rim and spokes. In this stage of the investigation, it was proposed that
stress and strains are caused by short-term loading, and the behavior of the material is
elastic, and conforms to Hooke's law. The shape that the rim assumes and other parameters
are determined by the plane axlsymmetrlc character of the loading, and also by the axisym-
metric anlsotropy, or quasi-isotropy of the material used. It was therefore assumed that
the flywheel has a plane median surface to which the load is applied. In this case, the
stressed state was considered plane. The loading consisted of a mass force caused by rota-
tion, and uniformly distributed radial boundary forces.
i. The equation of equilibrium for an elementary volume bounded by two radial sections
and two concentric circles placed at a distance r from the disk's axis of rotation assumes
the form (Fig. 2)
d__ (rot) __ o, + Pr = 0, (i)
dr
Institute of Strength Problems, Academy of Sciences of the Ukrainian SSR, Kiev. Trans-
lated from Problemy Prochnosti, No. 8, pp. 98-102, August, 1985. Original article submitted
February 27, 1984.
0039-2316/85/1708-1151509.50 �9 1986 Plenum Publishing Corporation 1151
Fig. i. Diagram showing spoked flywheel.
where P = p~ar; o r and o~ are the radial and circumferential stresses, p is the density of
the material, and ~ is the angular rotation frequency of the object.
A cerain relation exists between the deformations and displacements for the plane case
with consideration given to axial symmetry:
du u (2)
e r = ~ f ; e~ = - 7 - "
Making use of Hooke's law for an anisotropic material in the case of a plane stressed
state
o . o~ a~ o r
= ~ , -- ; ~ -- ( 3 ) Er %--%or EW -" E~ Vr~ E r'
and also the condition of interaction for the elastic constants
Er~wr = Ewvr ~, ( 4 )
we obtain the following differential equation for radial displacement:
where
dr 2d2u ~_ �9 dudr k ~ - ~ + qr ----. O, ( 5 )
k 2 ----- E~/E~; I - - %,~pr~r(p
q = Er P 02"
Equation (5) has the following solution:
u = C~r k + C~r - ~ q r3, (6) 9- - k 2
where the constants of integration C, and Ca are determined from equilibrium conditions near
the edges of the disk.
Expressions for the radial and circumferential stresses can be obtained from Eqs. (3)
and (6). Introducing the new constants c~ and c= and making use of condition of interac-
tion (4), we can write:
fir = cz r ~ - ] 4- 02 r - k - ~ - - 3 -t- V~r pto2r2;
9 - - k 2
( 7 )
~ = czk . rk - I ~ c2kr-l~-z k 2 (3vrq~ --[- | )
9 - - k z P ~
The relation between c~ and c2 and the constants of integration used in (6) is expressed
in the following manner:
E r (k + "Vwr ) C 1 . E r (Vcp r - - k) C 2
= , c2 = (8)
C l 1 - - ~ r q ~ r p r l - - Vrq~'Vq~ r
The problem for a disc with an opening free of perimeter loads can be solved in the
first stage of the computation of the stress--strain state of a spoked flywheel. Considering
the oonditions or(R) = or(re) = 0, we obtain
1152
3 -{- ~'_ r ( J~+3-- r~0+3 ~
(9)
3 + ~ , ~ ~[70 +3R 2* -Rk+ 3
c, = o= t '~
for the case in question.
Let us compute the values of o r and o~ with consideration Kiven to the relationships
derived for the constants c, and c2. For this purpose, let us introduce the dimensionless
quantities
^ ~ ^
r = riG; = R/r , ; r o = I.
T h e n , we c a n f i n a l l y w r i t e o r a n d o~0 i n t h e f o l l o w i n g m a n n e r :
~ ' - - 9----g~-~-l~176 ~ - - - q i 4 ~ - I \ r ) - - 1 ; (i0)
�9 fk-- . k - - ~2k 3"~%'~r " (11 )
In the case of radial displacements, we can derive an expression for u from (6):
3-1- Vir rR k + 3 - ~k--3-- I (R'~-~) k+3 (k -I- ~,r) - - ki--V'r'] E;~:~,) P = ' : L ~ ] l~'- ' (k-%') ~ - ] ,?, a+~, �9 u =
If the outer perimeter of the disk is not free of loads, i.e.,
OR~= O, Or(ro) = 0 are satisfied, we have from (7)
OR ~k+3 3 -'~ Vqw ^k+3
R~ - + -~-~-~- po= (R -- l)
c~ = (~2k_ l) ~0 -3
eR ~3-~ 3+V,ro=i (~3-h l)
R --i + 9 - - k ' " --
~ 3 + vor ^ ]
( ~ k l) (k + ripe)
C 2
Let us introduce the operator
A (k, r) =
(12)
the conditions or(R) =
(13)
(14)
(15)
from ex- Considering this designation, we obtain the relationship for displacements
pressions (6), (13), and (14):
r~(l--,,vr,) A(k,r) + A(__k,r)_r 3 po' ].
u = E ~-~-~i-j (16)
The stress distribution in the disc is described by the equations
{ ^1 ^ ^ I 3 + V~r p~2r2t ru; a, = [A (k, r) (k + v,D + A ( - - k, r) (v , , - - k)] r - o - - h' (17)
% = [ A ( k , r ~ ( k + v ~ r ) k __ A (__ k . r) (Vxr __ k) k] r - I k= (3Vr, + 1) ^ 2 9--k' po2: ro. (18)
2. Let us derive the equilibrium and deformation equations for the spokes. Let us
isolate in the spoke an elementary volume dv bounded by the radial planes of the flywheel
with angle of opening d~. Let us direct the ordinate of the XOY coordinate system along
the axis of the spoke toward the rim (Fig. 2b).
The centrifugal force acting on the isolated volume of the spoke is determined by the
relationship
dC= p~'rdo, (19)
where dv = Sdy and S is the cross-sectional area of the spoke.
1153
2
/
. -drhcl~
,-#fh
a
b
Fig. 2. Forces acting on small volume
of disk (a) and spoke (b) existing in
centrifugal-force field.
Let us write the relationships of certain quantities in rectangular coordinates in terms
of polar coordinates:
y = rsin ~ 2
Expressing r in terms of R, we obtain
T h u s )
dy = - ~ sin ~ ,in-~ (~ + ~)d~.
or
(21)
sin
d C = - - f ~ ' R ~ 2 sin' (~ +q~i . S d ~ , (22)
s,n-~ cos (~ + q~)
dCv = - - poJ'R 2 �9 Sdq~;
sin' (~ + q 0
s in ~
2
dC:~ = - - pco2R ' �9 Sdc~
sin' (-~-~ + ~p)
(23)
1154
!.
;.i
w I q '
-J2-:O ~ :3 22%.MPa 700 8&~ q00 10~ 'W ~2~ 3~.MPa
F i g . 3 . D i s t r i b u t i o n o f r a d i a l ( 1 , 3) and c i r c u m -
f e r e n t i a l (2, 4) stresses in rim of flywheel at ro-
tation frequency of 65,600 rpm (I), 70,800 rpm (2)
53,600 rpm (3), and 59,900 rpm (4). (Solid lines
denote rim of fiberglass flywheel, and broken lines
rim of carbon-fiber-reinforced-plastic flywheel.)
in projections on the coordinate axes.
In this stage of the computations, the effect of the components C x on the stress--strain
state of the spoke can be neglected in view of its smallness. The expression for the stress
in the cross section of the spoke due to ~ assumes the following form:
(24)
The integration of this expression yields the relationship
(25)
day =
The extremum of expression (25) can be determined from the condition-~- m 0; hence,
- - T
~ i.e. when The maximum The resultant condition is satisfied when nu~=-~ , ~max.
value of the stresses in the spoke is
Analysis indicates that cos2B/2 > 0.9 when R/r~ > 3. Consequently, the maximum stresses
in the inclined (B ~=0) spokes differ from those in the straight (8 = O) spokes by no more
than 10%. Here, this difference diminishes markedly with increasing ratio R/ro. Herein-
after, the assumption that the spokes are straight is therefore made to simplify computa-
tions in studying the stress~train state of flywheel elements. The equation of equilibrium
of a straight spoke (rod) in a centrifugal-force field can be written as
do r
d--7- + p~r = 0. (27)
After integrating and considering the condition or(R) = 0, we obtain
~, = p~2 (R2- - r2 ) , ' 2 . (28)
The spoke's strain e r is expressed in the following way:
f , ~ 2 ~ ~ (29) ~r = - ~ - ( R - - - r ) .
Let us find the displacement, assuming that u = 0 when r = 0:
i p~2f ~ r ~ (30) tt = e,dr = ~ ~ R - r - - T l .
X ~
0
1155
The displacement of the end of the spoke (r = R) is determined by the relationship
tt (R) - - P<~ ( 3 1 )
3E
3. The problem of acquiring the stress--strain state of all flywheel elements is sta-
tically indetermlnant. For its solution, let us make use of the compatibility of rim and
spoke deformations. The displacements of the rim and spokes on a radius R should be:
U~ (~) + U ~ (Q) = u ~) -- ud(Q), (32)
where Q is an unknown force acting along the rim of the disk owing to its interaction with
the spokes, uSP(m) is the displacement in a spoke due to the effect of centrifugal forces,
usp(Q) is the displacement in a spoke due to the effect of the deformed disk, ud(w) is the
displacement in the disk due to the effect of centrifugal forces, and ud(Q) is the displace-
ment in the disk due to the deformed spokes.
Let the spokes be fixed on the rim at m points; four spokes radiate from each point.
Stresses in the spokes generate a perimeter loading for the disk. The relation between the
stress through the rim of the disk OR d and the stress at the end of a spoke osP R is expressed
by the relationship
Q = -- o~h. 2~R = o~4mS, (33)
where h is the thlckness of the disk, and odR corresponds to o R in expressions (13)-(18).
Thus,
Oy = ~Rh
-- ~ an. (34)
Considering what we have stated, let us write the equation for determination of o R. The
right side of relationship (32) is expression (16) for r = R, where o R is taken with a
"minus" sign. Let us express the left side of relationship (32), considering expression
(31):
u~ (co) + u~ (Q) = p~2R' ~R~h~
3s 2 m S f S p �9 ( 3 5 )
Finally, we have
3 ~ RVtaR" I
4 (, [A (k, + A ( - - k, fi) - - 1 = ' (36)
"" 9 - - k 2 ] Es-'~ " ~ - 2mS ]" Ed
Having determined o R from expression (36), it is possible to calculate the distribution of
the stresses o r and o,~ in the disk from relationships (17) and (18).
4. Using this solution, we calculated the stress--strain and limiting state of a fiber-
glass flywheel, data on the geometric and mechanical parameters of which are presented in
[i]. According to Eq. (36), we obtain
oR = - - 0. 5 6 4 r 2. ( 3 7 )
We then computed the limiting rotational frequencies corresponding to spoke failure
~SPmax, rim delamlnation ~drmax , and rim failure ~d~max , as well as values of o R for these
cases. The numerical values of the desired parameters are as follows:
sp
COmax = 7 , 6 4 . 1 0 ~ rpm o" a = - - 36 .1 MPa
d
r 6 . 6 5 . I0* rpm oR = - - 2 7 , 4 MPa
o~dmax=7 .13 - 104 rpm a n = - - 3 1 . 4 MPa
The resultant data are close to the results of the above-described study [I]. Here,
we may also conclude that the bearing capacity of the flywheel is governed by the radial
strength of the rim. Comparison with experimental data should be considered more satis-
factory for the computation performed in the present study, however, since the minimum
limiting rotation frequency (5.84"10 ~m rpm) is lower in [i] than that attained during the
failure of a flywheel on an acceleration bench.
The stress distributed in the rim of the flywheel at limiting rotatinn frequencies as
computed from ~qs. (17) and (18) is shown in Fig. 3. Curve 1 (Fig. 3) corresponds to radial
stresses when ~drmax = 6.65.104 rpm. o r = --27.4 MPa is attained on the outer perimeter of
the rim (r = R = 1.4), and the largest tensile radial stresses o r = 27 MPa on the radius
1156
r = 1.15; this corresponds to the limiting strength of the composite in the direction per-
pendicular to the reinforcement. The distribution of circumferential stresses for a rota-
tion frequency ~d~max = 7.13'10 ~ rpm is shown in Fig. 3 (curve 2). Circumferential stresses
equal to the ultimate strength in the direction of the reinforcement (1200 MPa) are attained
on the radius of the internal perimeter r = ro = i.
5. The strength parameters of a compatible model of a flywheel formed from other com-
posite materials are computed in a similar manner.
The material of the flywheel rim is a carbon-fiber-reinforced plastic with 30% of
epoxy binder. The physicomechanical characteristics of the rim composite are as follows:
0 = 1.44 "lOs ks/m3, E~ = 1.3"105 MPa (in the circumferential direction), E r = 5400 MPa (in
the radial direction, LI~ = 1400 MPa (ultimate strength in the circumferential direction),
and II r = 14 MPa (ultimate strength in the radial direction). The mass of the rim is 830 g.
The flywheel has 52 spokes, which intersect the outer perimeter at 13 points. Each
spoke contains i0 braids of an organoplastic with 30% of epoxy binder. The overall mass of
the spokes is 550 g. The physicomechanical characteristics of the spoke composite, which
are required for the computation, are as follows: 0 = 1.31"103 kg/ms, E = 6.5"10 ~ MPa, and
II = 1800 MPa. The geometric parameters of the flywheel are as follows: R = 0.165 m, ro =
0.135 m, and h = 0.024 m.
Numerical calculation of the stress--strain state of the construction under investigation
produced the following results:
sp
c o ~ = 9 . 0 7 . 1 0 " rpm (FR = - - 19.8 MPa
d
~,max = 5- 82"10 ~ ~ m o R = - - 8 . 2 MPa
d
~ m , x = 6.06" I0 ~ rpm oR = - - 8 .9 MPa
The s t r e s s d i s t r i b u t i o n i n t h e c a r b o n - f i b e r - r e l n f o r c e d - p l a s t i c r i m o f t h e f l y w h e e l
u n d e r c o n s i d e r a t i o n , w h i c h c o r r e s p o n d s t o t h e l i m i t i n g r o t a t i o n f r e q u e n c i e s , i s shown i n
F i g . 3 ( r e l a t i o n s h i p s 3 and 4 ) : r a d i a l s t r e s s e s o r f o r ~d rmax = 5 . 8 2 ' 1 0 ~ rpm and c i r c u m -
f e r e n t i a l stresses o~ for ~d~max = 6.06"10 ~ rpm. Maximum radial stresses Ormax = 14 MPa
develop on the radius r = I.i.
Bench tests of the flywheel model that we have presented, which were performed by the
Zhitomir Branch of the Kiev Polytechnic Institute, suggest that its ultimate bearing capacity
is exhausted at a rotation frequency of 50,000 rpm. This can be considered to be in good
agreement with the results of computations, bearing in mind the statistical character of
the physicomechanical properties of composite materials.
i.
LITERATURE CITED
P. A. Moorlat and G. G. Portnov, "Computation of the stress--strain state of a chord
flywheel with spokes," Mekh. Kompozitn. Mater., No. 5, 853-862 (1983).
1157
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